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CONCEPT MAP GEOMETRY August 2011 Suggested Sequence: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. Tools of Geometry Reasoning and Proof Parallel and Perpendicular Lines Congruent Triangles Relationships Within Triangles Polygons and Quadrilaterals Similarity Right Triangles and Trigonometry Transformations Area Surface Area and Volume Circles GEOMETRY Prioritized Curriculum CSOs M.O.G.3.1 M.O.G.3.2 M.O.G.3.3 M.O.G.3.4 M.O.G.3.5 M.O.G.3.6 M.O.G.3.7 M.O.G.3.8 M.O.G.3.9 M.O.G.3.10 M.O.G.3.11 M.O.G.3.12 M.O.G.3.13 M.O.G.3.14 M.O.G.3.15 M.O.G.3.16 M.O.G.3.17 M.O.G.3.18 M.O.G.3.19 M.O.G.3.20 M.O.G.3.21 Essential X X X Important X X X X X X X X X X X X X X X X X X Need to Know Tools of Geometry Key Concepts: • Nets and Drawings for Visualizing Geometry • Points, Lines, and Planes • Measuring Segments • Measuring Angles • Exploring Angle Pairs • Basic Constructions • Midpoint and Distance in the Coordinate Plane • Perimeter, Circumference, and Area Key Vocabulary: CSOs: G.3.1, G.3.8, G.3.17, G.3.21 Estimated days to complete: 8 Enduring Understanding: Students will be introduced to various topics in the study of geometry. Essential Question(s): • How can you represent a three-dimensional figure with a two-dimensional drawing? • What are the building blocks of geometry? • How can you describe the attributes of a segment or angle? • • • • • • • • • • • • • • • • • • • • Acute, right, obtuse, straight angles Adjacent angles Angle bisector Collinear points, coplanar points Complementary and supplementary angles Congruent angles Congruent segments Constructions Linear pair Measure of an angle Net Perpendicular bisector Perpendicular lines Point, line, plane Postulate, axiom Ray, opposite rays Segment Segment bisector Vertex of an angle Vertical angles Reasoning and Proof Key Concepts: • Patterns and Inductive Reasoning • Conditional Statements • Biconditionals and Definitions • Deductive Reasoning • Reasoning in Algebra and Geometry • Proving Angles Congruent Key Vocabulary: CSOs: G.3.2, G.3.3, G.3.4, G.3.5, G.3.7 Estimated days to complete: 5 Enduring Understanding: Students will be introduced to topics related to reasoning including deductive and inductive reasoning. Essential Question(s): • How can you make a conjecture and prove that it is true? • • • • • • • • • • • • • • • • • • • • Biconditional Conclusion Conditional Conjecture Contrapositive Converse Counterexample Deductive reasoning Equivalent statements Hypothesis Inductive reasoning Inverse Law of detachment Law of syllogism Negation Paragraph proof Proof Theorem Truth value Two-column proof Key Concepts: • • • • • • • • Lines and Angles Properties of Parallel Lines Proving Lines Parallel Parallel and Perpendicular Lines Parallel Lines and Triangles Constructing Parallel and Perpendicular Lines Equations of Lines in the Coordinate Plane Slopes of Parallel and Perpendicular Parallel and Perpendicular Lines CSOs: G.3.1, G.3.4, G.3.5, G.3.6, G.3.14, G.3.19, G.3.20 Estimated days to complete: 8 Enduring Understanding: Students will expand upon their understanding of skills related to parallel and perpendicular lines. Essential Question(s): • How do you prove that two lines are parallel or perpendicular? • What is the sum of the measures of the angles of a triangle? • How do you write an equation of a line in the coordinate plane? Key Vocabulary: • • • • • • • • • • • • • • • Alternate exterior angles Alternate interior angles Auxiliary line Corresponding angles Exterior angle of a polygon Flow proof Parallel lines Parallel planes Point-slope form Remote interior angles Same-side interior angles Skew lines Slope Slope-intercept form Transversal Congruent Triangles Key Concepts: • Congruent Figures • Triangle Congruence by SSS and SAS • Triangle Congruence by ASA and AAS • Using Corresponding Parts of Congruent Triangles • Isosceles and Equilateral Triangles • Congruence in Right Triangles • Congruence in Overlapping Triangles Key Vocabulary: CSOs: G.3.5, G.3.7 • Estimated days to complete: 7 • • • • • • • Enduring Understanding: Students will build upon their understanding and skills related to angles and triangles. Essential Question(s): • How do you identify corresponding parts of congruent triangles? • How do you show that two triangles are congruent? • How can you tell whether a triangle is isosceles or equilateral? Base angles of an isosceles triangle Base of an isosceles triangle Congruent polygons Corollary Hypotenuse Legs of an isosceles triangle Legs of a right triangle Vertex angles of an isosceles triangle Key Concepts: • Midsegments of Triangles • Perpendicular and Angle Bisectors • Bisectors in Triangles • Medians and Altitudes • Indirect Proof • Inequalities in One Triangle • Inequalities in Two Triangles Relationships Within Triangles CSOs: G.3.5, G.3.10, G.3.18 Estimated days to complete: 6 Enduring Understanding: Students will identify the unique properties of triangles. Essential Question(s): • How do you use coordinate geometry to find relationships within triangles? • How do you solve problems that involve measurements of triangles? • How do you write indirect proofs? Key Vocabulary: • • • • • • • • • • • • • • • Altitude of a triangle Centroid of a triangle Circumcenter of a triangle Circumscribed about Concurrent Distance from a point to a line Equidistant Incenter of a triangle Indirect proof Indirect reasoning Inscribed Median of a triangle Midsegment of a triangle Orthocenter of a triangle Point of concurrency Polygons and Quadrilaterals Key Concepts: • The Polygon Angle-Sum Theorem • Properties of Parallelogram • Proving that a Quadrilateral is a Parallelogram • Properties of Rhombuses, Rectangles, and Squares • Conditions for Rhombuses, Rectangles, and Squares • Trapezoids and Kites • Polygons in the Coordinate Plane • Applying Coordinate Geometry • Proofs Using Coordinate Geometry Key Vocabulary: CSOs: G.3.4, G.3.5, G.3.8, G.3.14, G.3.17 Estimated days to complete: 8 Enduring Understanding: Students will identify the unique properties of triangles. Essential Question(s): • How do you use coordinate geometry to find relationships within triangles? • How do you solve problems that involve measurements of triangles? • How do you write indirect proofs? • • • • • • • • • • • • • • • Base, base angle, and leg of a trapezoid Consecutive angles Coordinate proof Equiangular, equilateral polygon Isosceles trapezoid Kits Midsegment of a trapezoid Opposite angles Opposite sides Parallelogram Rectangle Regular polygon Rhombus Square trapezoid Similarity Key Concepts: • • • • • Ratios and Proportions Similar Polygons Proving Triangles Similar Similarity in Right Triangles Proportions in Triangles Key Vocabulary: CSOs: G.3.4, G.3.5, G.3.9, G.3.11, G.3.19 Estimated days to complete: 5 Enduring Understanding: Students will expand upon their understanding and skills related to similarity. Essential Question(s): • How do you use proportions to find side lengths in similar polygons? • How do you show two triangles are similar? • How do you identify corresponding parts of similar triangles? • • • • • • • • • • • • Extended proportion Extended ratio Extremes Geometric mean Indirect measurement Means Proportion Ratio Scale drawings Scale factor Similar figures Similar polygons Key Concepts: • The Pythagorean Theorem and its Converse • Special Right Triangles • Trigonometry • Angles of Elevation and Depression • Vectors Right Triangles and Trigonometry CSOs: G.3.11, G.3.12 Estimated days to complete: 6 Enduring Understanding: Students will explore concepts related to right triangles, including trigonometry. Essential Question(s): • How do you find a side length or angle measure in a right triangle? • How do trigonometric ratios relate to similar right triangles? • What is a vector? Key Vocabulary: • • • • • • • • • • • • Angle of depression Angle of elevation Cosine Initial point Magnitude Pythagorean triple Resultant Sine Tangent Terminal point Trigonometric ratios Vector Transformations Key Concepts: • • • • • • Translations Reflections Rotations Symmetry Dilations Compositions of Reflections • Tessellations CSOs: G.3.15, G.3.19 Estimated days to complete: 6 Enduring Understanding: Students will explore the concepts related to transformations. Essential Question(s): • • • • How can you change a figure’s position without changing its size and shape? How can you change a figure’s size without changing its shape? How can you represent a transformation in the coordinate plane? How do you recognize symmetry in a figure? Key Vocabulary: • • • • • • • • • • • • • • • • Center of a regular polygon Composition of transformations Dilation Glide reflection Glide reflectional symmetry Image Isometry Line symmetry Preimage Reflection Rotation Rotational symmetry Tessellation Transformation Translation Translational symmetry Area Key Concepts: • Areas of Parallelograms and Triangles • Areas of Trapezoids, Rhombuses, and Kites • Areas of Regular Polygons • Perimeters and Areas of Similar Figures • Trigonometry and Area • Circles and Arcs • Areas of Circles and Sectors • Geometric Probability Key Vocabulary: CSOs: G.3.8, G.3.9, G.3.11, G.3.13 Estimated days to complete: 7 Enduring Understanding: Students will identify the unique properties of triangles. Essential Question(s): • How do you find the area of a polygon or find the circumference and area of a circle? • How do perimeters and areas of similar polygons compare? • • • • • • • • • • • • • • • • • • • • Adjacent arcs Altitude Apothem Arc length Base Central angle Circle Circumference Concentric circles Congruent arcs Congruent circles Diameter Geometric probability Height Major arc Minor arc Radius Sector of a circle Segment of a circle semicircle Surface Area and Volume Key Concepts: • Space Figures and Cross Sections • Surface Areas of Prisms and Cylinders • Surface Areas of Pyramids and Cones • Volumes of Prisms and Cylinders • Volumes of Pyramids and Cones • Surface Areas and Volumes of Spheres • Areas and Volumes of Similar Solids Key Vocabulary: CSOs: G.3.9, G.3.16 Estimated days to complete: 8 Enduring Understanding: Students will find surface areas and volumes of solid figures. Essential Question(s): • How can you determine the intersection of a solid and a plane? • How do you find the surface area and volume of a solid? • How do the surface areas and volumes of similar solids compare? • • • • • • • • • • • • • • • • • • • • Altitude Center of a sphere Cone Cross section Cylinder Edge Face great circle Hemisphere Lateral area Lateral face Polyhedron Prism Pyramid Right cone Right cylinder Right prism Slant height Sphere Surface area volume Circles Key Concepts: • • • • Tangent Lines Chords and Arcs Inscribed Angles Angle Measures and Segment Lengths • Circles in the Coordinate Plane • Locus: A Set of Points Key Vocabulary: CSOs: G.3.13 Estimated days to complete: 5 Enduring Understanding: Students will explore concepts related to circles. Essential Question(s): • How can you prove relationships between angles and arcs in a circle? • When lines intersect a circle, or within a circle, how do you find the measures of resulting angles, arcs, and segments? • How do you find the equation of a circle in the coordinate plane? • • • • • • • • Chord Inscribed angle Intercepted arc Locus Point of tangency Secant Standard form of an equation of a circle Tangent to a circle